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Properties

Label 20.0.58299958569...1056.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{18}$
Root discriminant $17.31$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:C_5$ (as 20T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 4, 0, 16, 0, 53, 0, 102, 0, 111, 0, 70, 0, 27, 0, 9, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 + 9*x^16 + 27*x^14 + 70*x^12 + 111*x^10 + 102*x^8 + 53*x^6 + 16*x^4 + 4*x^2 + 1)
 
gp: K = bnfinit(x^20 + 3*x^18 + 9*x^16 + 27*x^14 + 70*x^12 + 111*x^10 + 102*x^8 + 53*x^6 + 16*x^4 + 4*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} + 9 x^{16} + 27 x^{14} + 70 x^{12} + 111 x^{10} + 102 x^{8} + 53 x^{6} + 16 x^{4} + 4 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5829995856912430117421056=2^{20}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.31$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{2}{23} a^{12} + \frac{7}{23} a^{10} - \frac{7}{23} a^{8} - \frac{9}{23} a^{4} + \frac{1}{23} a^{2} + \frac{6}{23}$, $\frac{1}{23} a^{15} + \frac{2}{23} a^{13} + \frac{7}{23} a^{11} - \frac{7}{23} a^{9} - \frac{9}{23} a^{5} + \frac{1}{23} a^{3} + \frac{6}{23} a$, $\frac{1}{23} a^{16} + \frac{3}{23} a^{12} + \frac{2}{23} a^{10} - \frac{9}{23} a^{8} - \frac{9}{23} a^{6} - \frac{4}{23} a^{4} + \frac{4}{23} a^{2} + \frac{11}{23}$, $\frac{1}{23} a^{17} + \frac{3}{23} a^{13} + \frac{2}{23} a^{11} - \frac{9}{23} a^{9} - \frac{9}{23} a^{7} - \frac{4}{23} a^{5} + \frac{4}{23} a^{3} + \frac{11}{23} a$, $\frac{1}{23} a^{18} - \frac{4}{23} a^{12} - \frac{7}{23} a^{10} - \frac{11}{23} a^{8} - \frac{4}{23} a^{6} + \frac{8}{23} a^{4} + \frac{8}{23} a^{2} + \frac{5}{23}$, $\frac{1}{23} a^{19} - \frac{4}{23} a^{13} - \frac{7}{23} a^{11} - \frac{11}{23} a^{9} - \frac{4}{23} a^{7} + \frac{8}{23} a^{5} + \frac{8}{23} a^{3} + \frac{5}{23} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -a^{18} - \frac{63}{23} a^{16} - \frac{189}{23} a^{14} - \frac{567}{23} a^{12} - 63 a^{10} - \frac{2135}{23} a^{8} - \frac{1687}{23} a^{6} - \frac{623}{23} a^{4} - \frac{96}{23} a^{2} - \frac{33}{23} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 119970.990716 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 10.4.2414538435584.1, \(\Q(\zeta_{11})\), 10.6.219503494144.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$